Evaluate the integrals in Exercises 39–56.
41. ∫2y dy/(y²-25)

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Identify the integral to solve: \(\int \frac{2y}{y^{2} - 25} \, dy\).

Recognize that the denominator is a quadratic expression that can be factored as \(y^{2} - 25 = (y - 5)(y + 5)\), but before factoring, check if substitution is more straightforward.

Use substitution by letting \(u = y^{2} - 25\), then compute \(du = 2y \, dy\), which matches the numerator times \(dy\) exactly.

Rewrite the integral in terms of \(u\): \(\int \frac{2y}{y^{2} - 25} \, dy = \int \frac{1}{u} \, du\).

Integrate \(\int \frac{1}{u} \, du\) to get \(\ln|u| + C\), then substitute back \(u = y^{2} - 25\) to express the answer in terms of \(y\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration of Rational Functions

This involves integrating functions expressed as the ratio of two polynomials. Techniques often include algebraic manipulation such as factoring or partial fraction decomposition to simplify the integral into manageable parts.

Partial Fraction Decomposition

A method used to break down complex rational expressions into simpler fractions that are easier to integrate. It is especially useful when the denominator can be factored into linear or quadratic terms.

Substitution Method

A technique where a substitution is made to simplify the integral, often by letting a part of the integrand equal a new variable. This can transform the integral into a basic form that is straightforward to evaluate.

Evaluate the integrals in Exercises 39–56.41. ∫2y dy/(y²-25)